Phenomenological approach to eukaryotic chemotactic efficiency
Bo Hu,1 Danny Fuller,2 William F. Loomis,2 Herbert Levine,1 and Wouter-Jan Rappel1
1
Department of Physics and Center for Theoretical Biological Physics,
University of California, San Diego, La Jolla, CA 92093-0374
2
Department of Biology, University of California, San Diego, La Jolla, CA 92093-0319
(Dated: January 22, 2010)
Eukaryotic cells are capable of detecting small chemical gradients for a wide range of background
concentrations. Ultimately, fluctuations place a limit on gradient sensing and recent work has
focused on the role of stochastic receptor occupancy as one possible limiting factor. Here, we
use a phenomenological approach to add spontaneous motility fluctuations to receptor noise and
predict the directional statistics of eukaryotic chemotaxis. Specifically, an Itô diffusion equation
with direction-dependent multiplicative noise is developed and analytically studied. We show that
our approach can naturally accommodate recent experimental data for the chemotaxis of the social
amoeba Dictyostelium.
PACS numbers: 02.50.Le, 05.65.+b, 87.23.Ge, 87.23.Kg
I.
INTRODUCTION
During chemotaxis, cells direct their movement by
sensing chemical gradients. Chemotaxis plays an important role in various biological processes including fertilization, neuronal development, wound healing and cancer
metastasis [1, 2]. Unlike prokaryotes, eukaryotic cells use
a spatial measurement of the asymmetric distribution of
bound receptors following the external gradient. These
bound receptors trigger the activation of intracellular signaling pathways, eventually leading to the generation of
directional movement. Interestingly, eukaryotic cells are
able to direct their motion in shallow gradients. For example, the social amoeba Dictyostelium discoideum is
observed to crawl up a chemoattractant gradient when
the front-back difference in the concentration is around
1% [3, 4]. Furthermore, the cells respond even when the
average local concentration is well below the dissociation
constant Kd . These findings has led to significant interest from the biophysics community in understanding the
effect of noise on chemotaxis [5–12].
Studies to date have mainly focused on fluctuations in
gradient sensing, arising for example from stochastic receptor dynamics [6, 10]. However, there are other sources
of noise, as is evidenced by the fact that cells move randomly in the absence of directional information [13, 14].
The simplest approach to adding random motility to directional bias is to assume that the cell executes Brownian motion in a deterministic periodic potential [15];
however this approach fails to take into account uncertainties in evaluating gradient steepness and direction.
Thus, what is needed is a new model of directed cell
motility that takes into account both sources of fluctuations. Such a model should be able to relate the relative importance of gradient-determination noise versus
motility noise to testable predictions for the behavior of
chemotactic cells. Here, we propose a phenomenological,
stochastic differential equation (SDE) model for this purpose and compare its predictions with the available data.
As we will see, our approach naturally accounts for the
Gradient Direction
YM-1
YM Y1
Y2
Y3
Y4
Ligand
Receptor
k+
⎯⎯
→ R1
L + R0 ←⎯
⎯
k−
Kd =
k−
k+
FIG. 1: (Color online). Schematic representation of our
model. The forward rate k+ and backward rate k− determine the transition between the unoccupied R0 and occupied
R1 states. The dissociation constant Kd is the ligand concentration at which half the receptors are bound in equilibrium.
unusual shape of the directional probability distribution
for chemotaxing cells.
II.
MODEL
In our model, the cellular surface takes a circular geometry where N independent receptors are uniformly distributed (Fig. 1). We divide the surface into M sectors
such that the local ligand concentration near each sector is constant while the number of receptors in each
sector Nm = N/M is large enough for the continuous approximation. For Dictyostelium, a typical value
is N = 60000, and one may choose M = 300 and
Nm = 200. We define the gradient steepness parameter as p = CL0 dC
dr , where C0 is the average local ligand
concentration across the cell length L. Then, the local
concentration at the mth sector with angular position ϕm
is given by Cm = C0 [1 + p2 cos(ϕm − φ)] where φ speci-
2
FIG. 2: The average cell speed in the experiments of Ref. [4]
as a function of angle for p=5% (A) and p=10% (B). For each
gradient steepness, the speed was computed by tracking the
position of the centroid of 30-40 cells during 8 minutes. For
more experimental details, see Ref. [4].
fies the gradient direction. The time for receptor decorrelation is dominated by the individual receptor rate
τm = (k− + k+ Cm )−1 which is typically faster than the
process by which motion can be mechanically altered [16].
This allows us to use the white-noise approximation
in which the number of occupied receptors is Ym =
hYm i + ηm = Nm Cm /(Cm + Kd ) + ηm for m = 1, ..., M ,
with hηm (t)ηn (s)i ≈ Nm Cm Kd /(Cm + Kd )2 δ(t − s)δmn
[7, 17]. Thus, the receptor signal is decomposed into M
independent Gaussian random variables.
The simplest estimate a cell can make of the gradient is obtained by the following statistic: Z =
PM
PM
m=1 Ym cos ϕm + i
m=1 Ym sin ϕm ≡ z1 + iz2 . When
M is large, one can evaluate Z via replacing the sum by
an integral. For small gradients (p < 10%), the integrand
can be expanded around p and we find
pN C0 Kd eiφ
+ O(p3 ) ≡ νeiφ + O(p3 ), (1)
4(C0 + Kd )2
N C0 K d
2
hz1,2
i ≈
+ O(p2 ) ≡ σ 2 + O(p2 ). (2)
2(C0 + Kd )2
hZi ≈
It is easy to check that z1 and z2 are uncorrelated Gaussian random variables, with different means but the same
variance. This implies that Z is a complex Gaussian
variable which can be written in polar coordinates as
Z = ρeiψ . Here, ρ measures the degree of asymmetry in
the ligand-bound receptor distribution while the phase
ψ estimates the gradient direction φ. Mathematically, ρ
follows the Rice distribution and ψ takes a symmetric
unimodal circular distribution [18]. It was found that for
large signal-to-noise ratio (SNR= ν/σ & 3) both ρ and
ψ are asymptotically Gaussian [18]. Thus, ρ ≈ ν + ηρ
with hηρ (t)ηρ (s)i = σ 2 δ(t − s), and ψ ≈ φ + ηψ with
hηψ (t)ηψ (s)i = (σ 2 /ν 2 )δ(t − s). As an orthogonal transformation from the Cartesian coordinates, ηρ is independent of ηψ . For typical Dictyostelium values we find that
SNR> 3, consistent with the Gaussian approximation.
Cellular motion can be decomposed into two stochastic processes: speed and direction. Recent experimental
data in Dictyostelium [4] show that the average speed is
roughly identical for all directions. In these experiments,
cells were exposed to stable exponential chemoattractant
profiles using microfluidic devices. The centroid of each
cell in the field of view was automatically tracked every
5s for 100 frames. Cells that moved the furthest without
colliding with another cell were chosen for data analysis.
Ten to twenty-five such cells were collected in each experiment, giving thousands of data points for each particular
gradient. The average speed as a function of the angle is
plotted in Fig. 2 for two different gradient steepnesses.
From this figure, we can conclude that any possible correlation between angle and speed is smaller than can be
obtained from the data. Thus, we restrict ourselves to
the directional process, parametrized by the migration
angle θ(t), by assuming a uniform migration speed.
Let us temporarily suppose that cells have perfect
knowledge about the gradient direction φ. In general, the
equation of θ(t) can be written as dθ/dt = G(ρ, θ−φ)+η0
along with hη0 (t)η0 (s)i = σ02 δ(t − s). The incorporation
of the gradient-independent noise η0 above allows the cell
to perform a random walk in the absence of any gradient.
Note that unlike a recent study of random motility [13],
we do not include colored noise in our model. The function G(ρ, θ) must exhibit the symmetry properties [15]:
(1) 2π-periodicity, G(ρ, θ ± 2nπ) = G(ρ, θ) for any integer n; (2) polar symmetry, G(ρ, θ ± π) = −G(ρ, θ),
because the system’s polarity will be reversed under
the reversal of gradient direction; (3) reflection symmetry, G(ρ, −θ) = −G(ρ, θ), i.e. cells cannot distinguish between left and right with respect to the direction φ. The simplest form obeying these requirements is
G(ρ, θ − φ) = −β(ρ) sin(θ − φ). Of course, cells have only
imperfect knowledge of the gradient direction. The phase
variable ψ in our model serves as an unbiased estimator
of φ for the cell. Thus, we arrive at the phenomenological
Langevin equation
dθ/dt = −β(ρ) sin(θ − ψ) + η0 .
(3)
This phenomenological equation describes a chemotactic cell trying to align its movement with the estimated gradient direction ψ. The first term describes
a gradient-dependent restoring force which steers the
cell toward the gradient direction and which contains
gradient-dependent fluctuations (arising from receptorligand dynamics and downstream pathways) while the
second term represents gradient independent noise in the
random motility machinery.
Without loss of generality, we will set φ = 0. In
the small noise limit we can perform a Taylor expansion
dθ/dt ≈ −β(ν) sin θ + β 0 (ν) sin θηρ + β(ν) cos θηψ + η0 =
−β(ν) sin θ + ηtot , where the variance of the total noise is
2
2
hηtot
i ≡ σtot
= [σβ 0 (ν) sin θ]2 +[σβ−1 (ν) cos θ]2 +σ02 , (4)
with β 0 (ν) = ∂β(x)/∂x|x=ν and β−1 (ν) = β(ν)/ν. The
first term in Eq. (4) describes the fluctuations arising
from ρ, the amplitude of the gradient estimate, while the
second term describes the fluctuations in ψ, the direction of the gradient estimate. Both of them vary with
0.4
0.1
−0.5
0
θ/π
0.5
κ=0.5
0.5
κ=2.5
κ=2.0
κ=1.5
0.6
0.2
0.4
1
λ
D 0.8
CI
0.3
0
0
1
λ=0
λ=0.4
λ=0.6
λ=0.8
0.4
Ps (θ)
κ=1.0
0.2
C 0.5
κ=1.0
λ= ε
κ=0.5
0.2
0.1
0
−1
0.6
A
κ=2.5
λ= ε
κ=2.0
κ=1.5
−0.5
0
θ/π
0.5
1
0
0
0.5
λ
1
FIG. 3: The stationary distribution Ps (θ|φ = 0; κ = 1.5, λ)
for (A) b > 1 and (C) b < 1 as a function of the parameter
λ representing the ratio of direction-dependent noise to maximal total variance. The chemotaxis index (CI) vs. λ for (B)
b > 1 and (D) b < 1, with the dashed lines indicating where
the bifurcation occurs.
the instantaneous direction of motion θ(t) and therefore
are multiplicative fluctuations. Since the cell responds
to noise at its current position, the multiplicative noise
is defined with the Itô prescription. From the previous
expression it is clear that the ultimate directional de2
pendence in σtot
relies on the sign of β 0 (ν) − β−1 (ν).
To capture possible nonlinear input-output characteristics of downstream pathways, we assume that β(ν) has
a power-law functional form, i.e. β(ν) = aν b . We can
distinguish between two qualitatively different cases. In
the first, b > 1, the total variance is of the “ sin θ” type,
2
2
σtot
= σ02 + (σβ−1 )2 + (β 02 − β−1
)(σ sin θ)2 and is maximal at θ = π/2; in the second, b < 1, the variance is
described by a “ cos θ” type and is maximal at θ = 0. In
the special case b = 1, the total variance is reduced to
2
σtot
= σ02 + a2 σ 2 with no directional dependence.
III.
RESULTS
We first study the case of b > 1 where the restoring
force is ultra-sensitive to the input signal. Define the
2
direction-independent noise σ⊥
= σ02 + (σβ−1 )2 and the
2
2
direction-dependent noise σk = (β 02 − β−1
)σ 2 . Then the
previous Langevin equation can be approximated as the
Itô SDE
q
dθ
2 η(t),
= −β(ν) sin θ + (σk sin θ)2 + σ⊥
(5)
dt
where η(t) is white noise. One can solve the associated
Fokker-Planck equation on the interval [−π, π] by imposing periodic boundary conditions [19]. The stationary
B
1
0.6
1
0.8
0.8
0.6
0.4
0.4
0.2
−6
0.2
0
−4
log10(a) −2
C 0.6
0 100
50
0
2
σ0
0.06
0.4
0.03
0.2
0.7 0.8 0.9
0
0
p=5%
0.5
θ/π
a=a*
a=2.5a*
a=0.25a*
a=a*, σ2tot=σ20
0
D
angle distribution
0.8
0.2
0
−1
1
CI
0.3
CI
0.4
Ps (θ)
B
λ=0
λ=0.4
λ=0.6
λ=0.8
angle distribution
A 0.5
CI
3
1
0.02 0.04 0.06 0.08 0.1
p
0.06
0.6
0.03
0.4
0.7
0.2
0.8
0.9
p=10%
0
0
0.5
1
θ/π
FIG. 4: (Color online). (A) CI maximized by choosing appropriate b for different values of a and σ02 , given p = 5% and
C0 = Kd = 30nM . (B) CI vs. p for C0 = Kd = 30nM ,
σ02 = 2, b = 1, and a = a∗ , 2.5a∗ , and 0.25a∗ . The solid
line corresponds to a model neglecting receptor noise, i.e.
2
= σ02 . (C) Experimental direction distribution (symbols)
σtot
for p = 5%, along with a fit using our model (solid line) and
the CN distribution (dashed line). The error bars represent
the standard deviation in the data set. The least square fitting parameters are λ = 0.5866, κ = 1.0222. (D) Same as (C),
but now for p = 10% and with fitting parameters λ = 0.7566,
κ = 1.8580. The insets show the tails of the distributions.
distribution reads
Ps (θ) =
i
h
2
arctanh(λ
cos
θ)
Ω exp κ 1−λ
λ
1 − (λ cos θ)2
,
(6)
2
where κ = 2β/σ⊥
is a parameter that compares the
direction-biased force with q
the direction-independent
2 is the ratio of
noise and where λ = σk / σk2 + σ⊥
the direction dependent and maximal total variance.
The normalization parameter Ω can be determined numerically. For a vanishing λ, we recover the Circular Normal (CN) distribution: limλ→0 Ps (θ; κ, λ) =
exp(κ cos θ)/(2πI0 (κ)). For large λ, our stationary distribution will appreciably deviate from the CN distribution with a sharper peak at θ = 0 and heavier tails near
θ = ±π (see Fig. 3A). A physical interpretation of this
leptokurtic feature can be given by realizing that the first
term in the Langevin equation (Eq. (3)) functions as a
restoring force. Contrary to the CN case, the amplitude
of this force, β(ρ), is taken from a distribution. Thus,
large values can occur and correspond to large restoring
forces which lead to a sharper peak in Ps (θ). Using the
same argument, one can see that the occurrence of small
values in this distribution are responsible for the heavy
tails in Ps (θ).
We can further explore the extrema and directionality
4
1+x
of Ps (θ) by recalling that arctanh(x) = 21 ln( 1−x
) and
2
2
setting ≡ κ(1 − λ )/(2λ) = λβ(ν)/σk , after which we
obtain the alternative form Ps (θ) = Ω(1+λ cos θ)−1 /(1−
λ cos θ)+1 . The solution of ∂θ ln Ps = 0 contains five
possible roots in the interval [−π, π]: θ = 0, θ = ±π,
and θ = ± arccos(−/λ) which exist only if λ > . As
seen in Fig. 3A, Ps (θ = ±π) correspond to global minima if λ < , but become local maxima once λ > ,
in which case the new global minima are located at
θ = ± arccos(−/λ). This change of extrema occurs at
a critical point λc = (equivalent to σk2 = β) where
the direction-dependent noise becomes comparable to the
magnitude of the restoring force. Direct numerical simulations of the full model described by Eq. (3) indicate
that our analytical expression for Ps (θ) works well when
SNR> 3 and λ < , justifying the Gaussian noise approximation.
As is common in the experimental literature, we can
define an order parameter hcos θi, the Chemotaxis Index
(CI), which is a measure of the directionality of the angular distribution. In Fig. 3B, we plot the CI for different
values of κ using our model. The dashed line in Fig. 3B
corresponds to the CI with a value of κ that satisfies the
critical condition λ = or equivalently κ = 2λ2 /(1 − λ2 ).
To the right of this line the distribution shows a local
maximum at ±π. The slope of the CI is close to 0 over
a large range of values for λ. This can be understood
by realizing that the negative effect of the heavy tails on
the CI is partially compensated by the positive effect of
a sharpened peak.
Next we look at the case of b < 1. The corresponding
stationary distribution is
h
i
2
Ω exp κ 1−λ
λ arctan(λ cos θ)
Ps (θ) =
.
(7)
1 + (λ cos θ)2
2
2
with σ⊥
− β 02 )σ 2 in the
= σ02 + (σβ 0 )2 and σk2 = (β−1
definitions for κ and λ. The analysis proceeds as above
and we only highlight the differences. The angle distribution exhibits not only heavy tails but also an obtuse
peak for increasing values of λ (Fig. 3C). This can be understood since in this case the noise in ψ, i.e. the second
term in Eq. (4), dominates and reduces the accuracy in
directionality. Due to the Gaussian approximation, the
analytical probability distribution exhibits double peaks
at θ = ± arccos(−/λ) when λ > [20]. Also, the CI
decays quickly with increasing λ even before the bifurcation (Fig. 3D), suggesting that cells should operate
in a regime where b ≥ 1. Finally, for b = 1, the direc2
tional dependence in σtot
disappears and the stationary
distribution reduces to the CN distribution with shape
parameter κ̃ = 2aν/(σ02 + a2 σ 2 ). Obviously, for larger
values of the parameter a, the system is more sensitive
to the gradient, and more of the receptor noise is transmitted to the directional output.
For any given a and σ02 , there exists a unique value of b
that maximizes the CI (Fig. 4A). We found numerically
that the combination of a∗ = σ0 /σ and b∗ = 1 optimizes
the CI over the whole parameter space (a, b) given the
level of σ0 . The optimal pair (a∗ , b∗ ) corresponds to the
ridge of the CI surface in Fig 4A. In summary, the exponent b is critical for the chemotactic responses. The
optimal output requires b = 1, while b > 1 is suboptimal
and b < 1 leads to poor directionality. This can be understood by realizing that when restoring force β(ν) is a
nonlinear function (i.e. b 6= 1) of the receptor signal ν,
the associated sensor noise is modulated to be directiondependent, which leads to heavy tails in the angle distribution (Fig. 3). In other words, there is a significant
probability that cells may move in the wrong direction
and the chemotactic performance is not optimal. An important implication of our model is that the CI is always
lower than that predicted by models neglecting receptor
noise (see Fig. 4B). In fact, previous experimental data
indicates that when the gradient steepness increases the
saturating value of the CI is less than one [5, 9]. This
feature is not consistent with previous CN models but
may be successfully explained by our b > 1 model where
2
the total noise σtot
becomes an increasing function of p.
We can compare our model to the results of recent experiments in which Dictyostelium cells were exposed to
stable exponential chemoattractant profiles [4]. The data
allow us to compute directional distributions for different
gradient parameters. To eliminate the slight asymmetric
bias introduced by the flow in the microfluidic chambers
we collected the data into 20 bins according to their absolute deviation from the gradient direction, |θ−φ|. The resulting distribution are shown as symbols for p = 5% and
p = 10% in Fig. 4C and 4D, respectively. We have also
plotted as a solid line a least squares fit to the data using
our model (Eq. (6)). Our fit captures well the heavy
tails and the sharp peaks, features that the CN distribution, plotted as a dashed line, is unable to reproduce. A
goodness of fit analysis using a Pearson’s chi-square test
revealed that the CN distribution is rejected at a significance level of less than 1% while our model exhibits large
p-values [21]. The observed leptokurtic feature suggests
that Dictyostelium cells are operating close to the optimal regime with the exponent b estimated as slightly
larger than 1.
IV.
DISCUSSION
A key assumption in our model is that the chemotaxing cell experiences a relatively stable gradient steepness
(p = const) regardless of the cell’s location or moving
history. Thus, our model results can be directly compared to experiments in which the gradient is carefully
quantified and controlled. If the gradient steepness is
spatially varying, the chemotactic response would be heterogeneous in space, possibly obscuring the structure of
the angular distribution.
Although we focus here on the role of receptor noise,
it is easy to extend our model to include extra gradientdependent fluctuations, by adding them to the magni-
5
tude of the restoring force, β(ρ). Such fluctuations might
arise from noise in the downstream signal transduction
pathways. Future work will track cells for longer periods of time and test for the presence of long-lived cell
individuality and other fluctuations sources [22]. Should
the data indicate such memory effects, our model could
be extended to include quenched fluctuations in cell responses.
DF, WFL, HL and WJR were supported by NIH Grant
P01GM078586. BH acknowledges support from NSF
PHY-0822283. We thank Y.-H. Tu, R. J. Williams, J.
Wolf, and T. Hwa for useful discussions.
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